UA-UMO 2005 Problem 114

Find all monotone (not necessarily strictly) functions $f: \mathbb{R}^{+}_0\rightarrow \mathbb{R}$ such that:

$f(x+y)-f(x)-f(y)=f(xy+1)-f(xy)-f(1) \forall x,y \ge 0$ ;

$f(3)+3f(1)=3f(2)+f(0).$